In this paper, we propose two new VLSI architectures for computing the N-point discrete Fourier transform (DFT) and its inverse (IDFT) based on a radix-2 fast algorithm, where N is a power of two. The first part of this work presents a linear systolic array that requires log2 N complex multipliers and is able to provide a throughput of one transform sample per clock cycle. Compared with other related systolic designs based on direct computation or a radix-2 fast algorithm, the proposed one has the same throughput performance but involves less hardware complexity. This design is suitable for high-speed real-time applications, but it would not be easily realized in a single chip when N gets large. To balance the chip area and the processing speed, we further present a new reduced-complexity design for the DFT/IDFT computation. The alternative design is a memory-based architecture that consists of one complex multiplier, two complex adders, and some special memory units. The new design has the capability of computing one transform sample every log2 N+1 clock cycles on average. In comparison with the first design, the second design reaches a lower throughput with less hardware complexity. As N=512, the chip area required for the memory-based design is about 5742×5222 μm2, and the corresponding throughput can attain a rate as high as 4M transform samples per second under 0.6 μm CMOS technology. Such area-time performance makes this design very competitive for use in long-length DFT applications, such as asymmetric digital subscriber lines (ADSL) and orthogonal frequency-division multiplexing (OFDM) systems